%5E3-expand.jpeg "(2x+2)^3 Expand")
Expanding (2x+2)^3
Expanding a binomial raised to a power can be done using the Binomial Theorem or by repeated multiplication. Let's explore both methods to expand (2x+2)^3.
1. Binomial Theorem
The Binomial Theorem provides a general formula for expanding expressions of the form (a+b)^n. It states:
(a + b)^n = a^n + na^(n-1)b + (n(n-1)/2!)a^(n-2)b^2 + ... + (n(n-1)...(n-k+1)/k!)a^(n-k)b^k + ... + b^n
Applying this to our case, where a = 2x, b = 2, and n = 3:
(2x + 2)^3 = (2x)^3 + 3(2x)^2(2) + 3(2x)(2)^2 + (2)^3
Simplifying:
(2x + 2)^3 = 8x^3 + 24x^2 + 24x + 8
2. Repeated Multiplication
We can expand (2x+2)^3 by multiplying (2x+2) by itself three times:
(2x+2)^3 = (2x+2)(2x+2)(2x+2)
First, expand the first two terms:
(2x+2)(2x+2) = 4x^2 + 4x + 4x + 4 = 4x^2 + 8x + 4
Now, multiply the result by (2x+2):
(4x^2 + 8x + 4)(2x+2) = 8x^3 + 16x^2 + 8x + 8x^2 + 16x + 8
Finally, combine like terms:
(2x+2)^3 = 8x^3 + 24x^2 + 24x + 8
Conclusion
Both methods lead to the same answer: (2x+2)^3 = 8x^3 + 24x^2 + 24x + 8. While the Binomial Theorem provides a direct formula, repeated multiplication might be easier to understand for those new to the concept.